Problem 31 - Algebra 2

Question

Solve each equation by completing the square. \(x^{2}-4 x+1=0\)

Step-by-Step Solution

Answer

The solutions are \(x = 2 + \sqrt{3}\) and \(x = 2 - \sqrt{3}\).

Step 1: Move Constant Term to the Right

To complete the square, we start by moving the constant term to the right side of the equation. This gives us: \[x^2 - 4x = -1\]

Step 2: Find the Number to Complete the Square

Take half of the coefficient of the linear term (which is -4), square it, and then add it to both sides of the equation. Half of -4 is -2, and squaring it gives us 4. Add 4 to both sides to maintain equality:\[x^2 - 4x + 4 = -1 + 4\]

Step 3: Rewrite the Left Side as a Perfect Square

The left side of the equation is now a perfect square trinomial, which can be factored as:\[(x - 2)^2 = 3\]

Step 4: Solve for x by Taking the Square Root

Take the square root of both sides. Remember to consider both the positive and negative roots:\[x - 2 = \pm \sqrt{3}\]

Step 5: Isolate x

Solve for \(x\) by adding 2 to both sides of the equation:\[x = 2 \pm \sqrt{3}\]

Key Concepts

Quadratic Equations

Quadratic equations are any expression that can be rewritten in the form \(ax^2 + bx + c = 0\). They represent parabolic graphs and have various solving methods. The expression typically involves a squared variable, a linear variable, and a constant. Each part plays a crucial role:

• \(x^2\) is the quadratic term, determining the shape of the parabola.
• \(bx\) is the linear term, affecting the parabola's symmetry.
• \(c\) is the constant term, shifting the parabola up or down.

Quadratic equations can be solved using multiple techniques like factoring, using the quadratic formula, or completing the square.

Perfect Square Trinomial

A perfect square trinomial is an expression like \((x - a)^2\) that expands to a quadratic of the form \(x^2 - 2ax + a^2\). Here's why it's important:

• Simplifies quadratic expressions by creating a common factor.
• Helps in understanding the completion of the square method.
• Transforms equations into easily solvable forms.

To create a perfect square trinomial, take half of the linear coefficient, square it, and adjust the equation, as seen in our exercise: turn \(x^2 - 4x\) into \((x-2)^2\) by squaring -2, which results in 4.

Solving Equations

Solving equations means finding the values that satisfy the given mathematical statement. For quadratic equations, this typically involves finding the roots or solutions where \(ax^2 + bx + c = 0\). In the context of completing the square, follow these steps:

• Rearrange the equation to isolate the quadratic and linear terms.
• Complete the square to transform it into a perfect square trinomial.
• Use algebraic manipulations to solve, often involving taking square roots.

Once the equation is in the form \((x - a)^2 = b\), apply the square root and solve for the variable by moving terms across the equation as needed.

Algebraic Methods

Algebraic methods for solving equations involve manipulating algebraic expressions to simplify or solve them. In completing the square, the manipulation focuses on re-arranging a quadratic into a perfect square form.

• Prioritize balancing both sides of the equation to maintain equality.
• Intentionally modify terms to create perfect squares.
• Understand each manipulation step's underlying concept.

These methods build the foundation of solving more complex equations, enhancing problem-solving skills that can be transferred across various mathematical fields. The algebraic techniques seen in our step-by-step exercise demonstrate how deliberate algebraic actions lead to straightforward solutions for complex expressions.